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I'm estimating a GLM with a continuous variable $X_1$, a dummy variable $X_2$ and the interaction term $X_1*X_2$.

After my understanding, when interpreting the coefficient of $X_1$ you should assume the value of zero for $X_2$, i.e. also for the interaction term.

Now a colleague suggested that if I skipped one of the main effects, then the interaction would not be a real interaction, i.e. I could simply interpret the coefficients of both - the remaining main effect and interaction term as if there was no interaction term in the model. I could not find any literature that supported this view. Is it correct?

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    $\begingroup$ So, the model is $g(\mu) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2$? I think the reasonable thing to do would be to interpret the effect of $X_1$ conditional on $X_2$, i.e. look at $\beta_1$ for $X_2 = 0$ and $\beta_1 + \beta_3$ for $X_2 = 1$. Essentially, interpret the regression separately for $X_2 = 0$ and $1$. $\endgroup$ – guy Aug 13 '12 at 4:59
  • $\begingroup$ @guy, that's basically the answer here. Would you care to re-post you comment as an answer (possibly w/ a little elaboration)? $\endgroup$ – gung - Reinstate Monica Aug 13 '12 at 16:12
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So, the model is $g(\mu) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2$. I think the natural thing to do would be to interpret the effect of $X_1$ conditional on $X_2$, i.e. look at $\beta_1$ for $X_2 = 0$ and $\beta_1 + \beta_3$ for $X_2 = 1$. This is tantamount to fitting a separate regression for $X_2 = 0$ and $X_2 = 1$; we have $g(\mu) = \beta_0 + \beta_1 X_1$ and $g(\mu) = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) X_1$ respectively.

I would say $X_1 X_2$ is as pure an interaction term as there is, given that you are using a function linear in $X_1$. If $\beta_3 = 0$, this says that the effect of a unit increase in $X_1$ is the same regardless of whether $X_2 = 1$ or $X_2 = 0$. Things get hairier as you start adding more predictors since the number of potential interaction terms grows exponentially; without some simplifying assumptions (e.g. a limit on the order and nature of the interactions) things can get unmanageable from a meaningful interpretation standpoint pretty fast.

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  • $\begingroup$ Thanks for the answers. I originally had the model as g(μ)=β 0 +β 1 X 1 +β 2 X 2 +β 3 X 1 X 2 (with some other covariates in the model) - however, my colleague thought that skipping β 2 X 2 would turn the interaction term β 3 X 1 X 2 into a "normal variable", i.e. I would not need to interpret β 1 X 1 conditional on β 3 X 1 X 2 anymore. I don t understand why this should be the case. $\endgroup$ – user13248 Aug 13 '12 at 21:31
  • $\begingroup$ That would be an odd thing to do I think. Usually one wouldn't remove a main effect while retaining an interaction term without having a good reason. Given that you want to do it that way, though, you can still interpret things by stratifying on $X_2 = 0$ and $X_2 = 1$. The coefficient on $X_1 X_2$ still represents the interaction between $X_1X_2$, but you are imposing the restriction that $g(\mu) = \beta_0$ when $X_1 = 0$, regardless of the value of $X_2$; that is, you are restricting the intercept of the two regressions to be the same. I guess one might call this a "shared intercept" model. $\endgroup$ – guy Aug 13 '12 at 22:20
  • $\begingroup$ Thanks for your quick reply. Can imposing the restriction "g(μ)=β 0, when X 1=0 , regardless of the value of X2" make sense, if we assumed that X2 itself had no relevant effect on the outcome variable? Even despite the restriction imposed, we do treat the interaction as "real" interaction (all interpretations as usual), is that right? Thanks, I really appreciate your help. $\endgroup$ – user13248 Aug 13 '12 at 22:55
  • $\begingroup$ If $X_2$ "had no relevant effect" then it ought not be in the model at all. $X_2$ does have a relevant effect if it is changing the influence of $X_1$. I'm really not sure why you want to do this. Even if $\beta_2$ is insignificant, if the interaction is significant it is standard practice to leave $\beta_2$ in. It only makes sense to me to drop $\beta_2$ if you have subject-matter knowledge that $X_2$ should not influence $g(\mu)$ at the value $X_1 = 0$, or maybe that $X_2$ only influences the outcome by magnifying/shrinking the effect of $X_1$, but even then I think you only stand to lose. $\endgroup$ – guy Aug 14 '12 at 1:48

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